There is a modern "Noahide" movement(I believe propagated by the Lubavitcher Rebbe), where people join together to be b'nei noah and not part of another religion. From some googling, I've seen that there are a fair amount of people on the web who identify religiously as Noahides. My question is, given that the 7 Noahide laws are relatively easy to follow (even if there are of course issues in Noahide halachah), what exactly do members of the modern Noahide movement do? For example, do they pray in an organized manner? If so, what forms do the prayers take? Do they study Torah? (Are they allowed to study Torah?) Or do they just follow the 7 rules? Is there any sort of organized religion for Noahides?
Writing as a ben Noach, there is no definitive answer on what we do; only what we can't do(save setting up a court). Bnei Noach can do whatever they want as long as they don't transgress the seven. Brit Noach is a covenant of wild freedom, basically. For the details of the seven, I go by the Rambam, Hilchot Melachim Ch. 8-10.
Since Brit Noach allows so much freedom, this doesn't mean you should be obnoxious, etc. particularly when living in a Jewish community. I don't put on tefillin, affix mezuzot, immerse in a mikvah, write Torah scrolls.
My wife and I, although we are not commanded to, we do Jew-ish things either to fit into the community or just because we want to. I pray from the Siddur Tehillat haShem(nusach Arizal), go to Shabbat services, daven Shachrit, give tzedakah, study Rambam, Chassidut, study Hebrew, learn with a Rabbi, teach other bnei Noach, go to work, etc.
Some bnei Noach just keep the seven and want to be left alone. Some want to be like gerim toshavim(although gerim toshavim are not halachically accepted today).
Brit Noach is instituted to give the human soul freedom in how deep he wants to go with haShem without being bad in this physical world, basically.
actually belief in G-d is part of those 7 commandments.
here are the 7 listed and defined (copied and pasted from chabad.org)
These are the Seven Noahide Laws, as enumerated in the Babylonian Talmud, Sanhedrin 56a:
1:carry out justice - An imperative to pursue and enforce social justice, and a prohibition of any miscarriage of justice.
2:no blasphemy - Prohibits a curse directed at the Supreme Being.
3: no idolatry - Prohibits the worship of any human or any created thing. Also prohibited is the making of idols and involvement with the occult. This necessitates an understanding of the One G-d of Israel and His nature.
4:no illicit intercourse - Prohibits adultery, incest, homosexual intercourse and bestiality, according to Torah definitions.
5:no homicide - Prohibits murder and suicide. Causing injury is also forbidden.
6:no theft - Prohibits the wrongful taking of another's goods.
7:don't eat a limb of a living creature - Promotes the kind treatment of animal life. It also encourages an appreciation for all kinds of life and respect for nature as G-d's creation.
furthermore in addition to these 7 commandments there are things a Noahide should be encouraged to partake in including prayer (actually the no blasphemy is written as a positive to bless but the inverse is also understood from that not to curse either), charity, modesty, visiting the sick, honoring ones parents, and other good deeds which bring honor to G-d and peace between man kind. A good website to find out more about the noahide movement is noahidenations.com
They definitely must believe in Hashem and keep His mitzvos as His commandments as apposed to thinking they are 'good ideas'. This is all in the rambam as others mentioned. There is a seffer called Mitzvos Hashem which deals extensively which mitzvos they are commanded in. There are alot more than seven, seven is only the number of mitzvos they get killed biyidei adam for transgressing. We know they are chayav missa bidei shomayim for things like keeping shabbos and learning torah, but those are not on the list of seven because those are not chayav bidei adam.